metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).12D10, C8⋊C22⋊2D5, (C5×D4).11D4, C4.178(D4×D5), C5⋊2C8.46D4, (C5×Q8).11D4, D4⋊D10⋊4C2, C4○D4.24D10, (C2×D4).79D10, C20.195(C2×D4), C20.D4⋊9C2, C5⋊5(D4.4D4), D4.4(C5⋊D4), D4.Dic5⋊5C2, Q8.4(C5⋊D4), C20.53D4⋊9C2, (C2×C20).14C23, C20.46D4⋊10C2, C10.124(C4⋊D4), (C2×D20).133C22, (D4×C10).104C22, C4.Dic5.24C22, C2.30(Dic5⋊D4), C22.13(D4⋊2D5), (C5×M4(2)).22C22, (C2×D4⋊D5)⋊22C2, (C5×C8⋊C22)⋊6C2, C4.51(C2×C5⋊D4), (C2×C4).14(C22×D5), (C2×C10).36(C4○D4), (C5×C4○D4).12C22, (C2×C5⋊2C8).170C22, SmallGroup(320,826)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).D10
G = < a,b,c,d | a8=b2=c10=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >
Subgroups: 446 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C5⋊2C8, C5⋊2C8, C40, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4.4D4, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4.Dic5, D4⋊D5, Q8⋊D5, C5×M4(2), C5×D8, C5×SD16, C2×D20, D4×C10, C5×C4○D4, C20.53D4, C20.46D4, C20.D4, C2×D4⋊D5, D4.Dic5, D4⋊D10, C5×C8⋊C22, M4(2).D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.4D4, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊D4, M4(2).D10
►On 80 points
(1 17 40 21 63 41 59 78)(2 79 60 42 64 22 31 18)(3 19 32 23 65 43 51 80)(4 71 52 44 66 24 33 20)(5 11 34 25 67 45 53 72)(6 73 54 46 68 26 35 12)(7 13 36 27 69 47 55 74)(8 75 56 48 70 28 37 14)(9 15 38 29 61 49 57 76)(10 77 58 50 62 30 39 16)
(1 63)(3 65)(5 67)(7 69)(9 61)(12 46)(14 48)(16 50)(18 42)(20 44)(22 79)(24 71)(26 73)(28 75)(30 77)(32 51)(34 53)(36 55)(38 57)(40 59)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 10 59 39 63 62 40 58)(2 57 31 61 64 38 60 9)(3 8 51 37 65 70 32 56)(4 55 33 69 66 36 52 7)(5 6 53 35 67 68 34 54)(11 12 72 26 45 46 25 73)(13 20 74 24 47 44 27 71)(14 80 28 43 48 23 75 19)(15 18 76 22 49 42 29 79)(16 78 30 41 50 21 77 17)
G:=sub<Sym(80)| (1,17,40,21,63,41,59,78)(2,79,60,42,64,22,31,18)(3,19,32,23,65,43,51,80)(4,71,52,44,66,24,33,20)(5,11,34,25,67,45,53,72)(6,73,54,46,68,26,35,12)(7,13,36,27,69,47,55,74)(8,75,56,48,70,28,37,14)(9,15,38,29,61,49,57,76)(10,77,58,50,62,30,39,16), (1,63)(3,65)(5,67)(7,69)(9,61)(12,46)(14,48)(16,50)(18,42)(20,44)(22,79)(24,71)(26,73)(28,75)(30,77)(32,51)(34,53)(36,55)(38,57)(40,59), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10,59,39,63,62,40,58)(2,57,31,61,64,38,60,9)(3,8,51,37,65,70,32,56)(4,55,33,69,66,36,52,7)(5,6,53,35,67,68,34,54)(11,12,72,26,45,46,25,73)(13,20,74,24,47,44,27,71)(14,80,28,43,48,23,75,19)(15,18,76,22,49,42,29,79)(16,78,30,41,50,21,77,17)>;
G:=Group( (1,17,40,21,63,41,59,78)(2,79,60,42,64,22,31,18)(3,19,32,23,65,43,51,80)(4,71,52,44,66,24,33,20)(5,11,34,25,67,45,53,72)(6,73,54,46,68,26,35,12)(7,13,36,27,69,47,55,74)(8,75,56,48,70,28,37,14)(9,15,38,29,61,49,57,76)(10,77,58,50,62,30,39,16), (1,63)(3,65)(5,67)(7,69)(9,61)(12,46)(14,48)(16,50)(18,42)(20,44)(22,79)(24,71)(26,73)(28,75)(30,77)(32,51)(34,53)(36,55)(38,57)(40,59), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10,59,39,63,62,40,58)(2,57,31,61,64,38,60,9)(3,8,51,37,65,70,32,56)(4,55,33,69,66,36,52,7)(5,6,53,35,67,68,34,54)(11,12,72,26,45,46,25,73)(13,20,74,24,47,44,27,71)(14,80,28,43,48,23,75,19)(15,18,76,22,49,42,29,79)(16,78,30,41,50,21,77,17) );
G=PermutationGroup([[(1,17,40,21,63,41,59,78),(2,79,60,42,64,22,31,18),(3,19,32,23,65,43,51,80),(4,71,52,44,66,24,33,20),(5,11,34,25,67,45,53,72),(6,73,54,46,68,26,35,12),(7,13,36,27,69,47,55,74),(8,75,56,48,70,28,37,14),(9,15,38,29,61,49,57,76),(10,77,58,50,62,30,39,16)], [(1,63),(3,65),(5,67),(7,69),(9,61),(12,46),(14,48),(16,50),(18,42),(20,44),(22,79),(24,71),(26,73),(28,75),(30,77),(32,51),(34,53),(36,55),(38,57),(40,59)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,10,59,39,63,62,40,58),(2,57,31,61,64,38,60,9),(3,8,51,37,65,70,32,56),(4,55,33,69,66,36,52,7),(5,6,53,35,67,68,34,54),(11,12,72,26,45,46,25,73),(13,20,74,24,47,44,27,71),(14,80,28,43,48,23,75,19),(15,18,76,22,49,42,29,79),(16,78,30,41,50,21,77,17)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | 10D | 10E | ··· | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 8 | 40 | 2 | 2 | 4 | 2 | 2 | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | D4.4D4 | D4×D5 | D4⋊2D5 | M4(2).D10 |
| kernel | M4(2).D10 | C20.53D4 | C20.46D4 | C20.D4 | C2×D4⋊D5 | D4.Dic5 | D4⋊D10 | C5×C8⋊C22 | C5⋊2C8 | C5×D4 | C5×Q8 | C8⋊C22 | C2×C10 | M4(2) | C2×D4 | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of M4(2).D10 ►in GL8(𝔽41)
| 21 | 3 | 32 | 11 | 0 | 0 | 0 | 0 |
| 15 | 6 | 2 | 2 | 0 | 0 | 0 | 0 |
| 37 | 2 | 38 | 38 | 0 | 0 | 0 | 0 |
| 27 | 39 | 3 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 28 | 0 | 15 |
| 0 | 0 | 0 | 0 | 2 | 28 | 17 | 15 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 37 | 12 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 0 | 1 |
| 18 | 40 | 17 | 24 | 0 | 0 | 0 | 0 |
| 36 | 23 | 38 | 0 | 0 | 0 | 0 | 0 |
| 12 | 16 | 17 | 1 | 0 | 0 | 0 | 0 |
| 29 | 28 | 40 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 34 | 25 | 1 | 9 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 40 | 39 | 16 |
| 20 | 24 | 24 | 20 | 0 | 0 | 0 | 0 |
| 26 | 21 | 3 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 25 | 1 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 33 | 39 | 7 |
G:=sub<GL(8,GF(41))| [21,15,37,27,0,0,0,0,3,6,2,39,0,0,0,0,32,2,38,3,0,0,0,0,11,2,38,17,0,0,0,0,0,0,0,0,2,2,12,37,0,0,0,0,28,28,12,37,0,0,0,0,0,17,0,12,0,0,0,0,15,15,0,11],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,3,0,0,0,0,0,40,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[18,36,12,29,0,0,0,0,40,23,16,28,0,0,0,0,17,38,17,40,0,0,0,0,24,0,1,24,0,0,0,0,0,0,0,0,0,34,1,33,0,0,0,0,0,25,0,40,0,0,0,0,1,1,0,39,0,0,0,0,0,9,0,16],[20,26,0,0,0,0,0,0,24,21,0,0,0,0,0,0,24,3,24,1,0,0,0,0,20,20,38,17,0,0,0,0,0,0,0,0,34,0,1,40,0,0,0,0,25,0,0,33,0,0,0,0,1,1,0,39,0,0,0,0,9,0,0,7] >;
M4(2).D10 in GAP, Magma, Sage, TeX
M_4(2).D_{10} % in TeX
G:=Group("M4(2).D10"); // GroupNames label
G:=SmallGroup(320,826);
// by ID
G=gap.SmallGroup(320,826);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,1123,297,136,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations